Proving The Maximum Area of a Shape Regardless of How Many Sides It Has

The greatest area of a quadrilateral given a fixed perimeter is a square. This seems to be true for an equilateral triangle. Is there a proof that given a fixed perimeter and fixed amount of sides, the area will be maximized when all the sides are equal regardless of how many sides there are?

SDK

December 30th, 2017 06:00 PM

I'm a bit short on time atm but here is the idea. Suppose a $n$-polygon is specified as a list of $n+1$ vertices, $\{(x_0,y_0),\dotsc,(x_n,y_n)\}$ and let $\gamma_k$ denote the line segment between the $(k-1)^{\rm st}$ and $k^{\rm th}$ vertex. Then the area is given explicitly by the line integral
\[\sum_{k=1}^n \int_{\gamma_k} x \ dy \]
Now, regard this as a function of $2(n+1)$-many variables and maximize it using standard multi-variable optimization techniques.

mathman

December 31st, 2017 01:46 PM

For any figure more than 3 sides, it is important to include a requirement that all angles be equal.
Example for 4 sides: a rhombus has all sides equal, but it can be squeezed to an area as close to 0 as one wants.